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Theorem selberg2lem 25039
 Description: Lemma for selberg2 25040. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Assertion
Ref Expression
selberg2lem (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
Distinct variable group:   𝑥,𝑛

Proof of Theorem selberg2lem
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 rpre 11715 . . . . . . . . 9 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2 chpcl 24650 . . . . . . . . 9 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
31, 2syl 17 . . . . . . . 8 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
43recnd 9947 . . . . . . 7 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
5 rprege0 11723 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6 flge0nn0 12483 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
75, 6syl 17 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℕ0)
8 nn0p1nn 11209 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℕ0 → ((⌊‘𝑥) + 1) ∈ ℕ)
97, 8syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℕ)
109nnrpd 11746 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℝ+)
1110relogcld 24173 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
1211recnd 9947 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13 relogcl 24126 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1413recnd 9947 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
1512, 14subcld 10271 . . . . . . 7 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
164, 15mulcld 9939 . . . . . 6 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17 fzfid 12634 . . . . . . 7 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
18 elfznn 12241 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1918adantl 481 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2019nnrpd 11746 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
21 1rp 11712 . . . . . . . . . . . . 13 1 ∈ ℝ+
22 rpaddcl 11730 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑛 + 1) ∈ ℝ+)
2321, 22mpan2 703 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 + 1) ∈ ℝ+)
2423relogcld 24173 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘(𝑛 + 1)) ∈ ℝ)
25 relogcl 24126 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
2624, 25resubcld 10337 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27 rpre 11715 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℝ)
28 chpcl 24650 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
2927, 28syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℝ)
3026, 29remulcld 9949 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
3130recnd 9947 . . . . . . . 8 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
3220, 31syl 17 . . . . . . 7 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
3317, 32fsumcl 14311 . . . . . 6 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34 rpcnne0 11726 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35 divsubdir 10600 . . . . . 6 ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ ∧ Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
3616, 33, 34, 35syl3anc 1318 . . . . 5 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
374, 12mulcld 9939 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
384, 14mulcld 9939 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
3937, 38, 33sub32d 10303 . . . . . . 7 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
404, 12, 14subdid 10365 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
4140oveq1d 6564 . . . . . . 7 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
42 fveq2 6103 . . . . . . . . . . 11 (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43 oveq1 6556 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
4443fveq2d 6107 . . . . . . . . . . 11 (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
4542, 44jca 553 . . . . . . . . . 10 (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
46 fveq2 6103 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
47 oveq1 6556 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
4847fveq2d 6107 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
4946, 48jca 553 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
50 fveq2 6103 . . . . . . . . . . . 12 (𝑚 = 1 → (log‘𝑚) = (log‘1))
51 log1 24136 . . . . . . . . . . . 12 (log‘1) = 0
5250, 51syl6eq 2660 . . . . . . . . . . 11 (𝑚 = 1 → (log‘𝑚) = 0)
53 oveq1 6556 . . . . . . . . . . . . . 14 (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
54 1m1e0 10966 . . . . . . . . . . . . . 14 (1 − 1) = 0
5553, 54syl6eq 2660 . . . . . . . . . . . . 13 (𝑚 = 1 → (𝑚 − 1) = 0)
5655fveq2d 6107 . . . . . . . . . . . 12 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
57 2pos 10989 . . . . . . . . . . . . 13 0 < 2
58 0re 9919 . . . . . . . . . . . . . 14 0 ∈ ℝ
59 chpeq0 24733 . . . . . . . . . . . . . 14 (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
6058, 59ax-mp 5 . . . . . . . . . . . . 13 ((ψ‘0) = 0 ↔ 0 < 2)
6157, 60mpbir 220 . . . . . . . . . . . 12 (ψ‘0) = 0
6256, 61syl6eq 2660 . . . . . . . . . . 11 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
6352, 62jca 553 . . . . . . . . . 10 (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
64 fveq2 6103 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
65 oveq1 6556 . . . . . . . . . . . 12 (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
6665fveq2d 6107 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
6764, 66jca 553 . . . . . . . . . 10 (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
68 nnuz 11599 . . . . . . . . . . 11 ℕ = (ℤ‘1)
699, 68syl6eleq 2698 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ (ℤ‘1))
70 elfznn 12241 . . . . . . . . . . . . . 14 (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
7170adantl 481 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
7271nnrpd 11746 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ+)
7372relogcld 24173 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
7473recnd 9947 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
7571nnred 10912 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
76 peano2rem 10227 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
7775, 76syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
78 chpcl 24650 . . . . . . . . . . . 12 ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
7977, 78syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
8079recnd 9947 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
8145, 49, 63, 67, 69, 74, 80fsumparts 14379 . . . . . . . . 9 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))))
827nn0zd 11356 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℤ)
83 fzval3 12404 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8482, 83syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8584eqcomd 2616 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
8619nncnd 10913 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
87 ax-1cn 9873 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
88 pncan 10166 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
8986, 87, 88sylancl 693 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
90 npcan 10169 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
9186, 87, 90sylancl 693 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
9289, 91eqtr4d 2647 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
9392fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
94 nnm1nn0 11211 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
9519, 94syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ0)
96 chpp1 24681 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) ∈ ℕ0 → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
9795, 96syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
9891fveq2d 6107 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
9998oveq2d 6565 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
10093, 97, 993eqtrd 2648 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
101100oveq1d 6564 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))))
10295nn0red 11229 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
103 chpcl 24650 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
104102, 103syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
105104recnd 9947 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
106 vmacl 24644 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
10719, 106syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
108107recnd 9947 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
109105, 108pncan2d 10273 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
110101, 109eqtrd 2644 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
111110oveq2d 6565 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
11220relogcld 24173 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
113112recnd 9947 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
114108, 113mulcomd 9940 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
115111, 114eqtr4d 2647 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
11685, 115sumeq12rdv 14285 . . . . . . . . 9 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)))
1177nn0cnd 11230 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℂ)
118 pncan 10166 . . . . . . . . . . . . . . . . 17 (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
119117, 87, 118sylancl 693 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
120119fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
121 chpfl 24676 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
1221, 121syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
123120, 122eqtrd 2644 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
124123oveq2d 6565 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
12512, 4mulcomd 9940 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
126124, 125eqtrd 2644 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127 0cn 9911 . . . . . . . . . . . . . 14 0 ∈ ℂ
128127mul01i 10105 . . . . . . . . . . . . 13 (0 · 0) = 0
129128a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (0 · 0) = 0)
130126, 129oveq12d 6567 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
13137subid1d 10260 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
132130, 131eqtrd 2644 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
13389fveq2d 6107 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
134133oveq2d 6565 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
13585, 134sumeq12rdv 14285 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
136132, 135oveq12d 6567 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
13781, 116, 1363eqtr3d 2652 . . . . . . . 8 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
138137oveq1d 6564 . . . . . . 7 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
13939, 41, 1383eqtr4d 2654 . . . . . 6 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))))
140139oveq1d 6564 . . . . 5 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
141 div23 10583 . . . . . . 7 (((ψ‘𝑥) ∈ ℂ ∧ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
1424, 15, 34, 141syl3anc 1318 . . . . . 6 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
143142oveq1d 6564 . . . . 5 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
14436, 140, 1433eqtr3rd 2653 . . . 4 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
145144mpteq2ia 4668 . . 3 (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
146 ovex 6577 . . . . 5 (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V
147146a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
148 ovex 6577 . . . . 5 𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V
149148a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
150 reex 9906 . . . . . . . 8 ℝ ∈ V
151 rpssre 11719 . . . . . . . 8 + ⊆ ℝ
152150, 151ssexi 4731 . . . . . . 7 + ∈ V
153152a1i 11 . . . . . 6 (⊤ → ℝ+ ∈ V)
154 ovex 6577 . . . . . . 7 ((ψ‘𝑥) / 𝑥) ∈ V
155154a1i 11 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ V)
15615adantl 481 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
157 eqidd 2611 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)))
158 eqidd 2611 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
159153, 155, 156, 157, 158offval2 6812 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘𝑓 · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))))
160 chpo1ub 24969 . . . . . 6 (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
161 0red 9920 . . . . . . . 8 (⊤ → 0 ∈ ℝ)
162 1red 9934 . . . . . . . 8 (⊤ → 1 ∈ ℝ)
163 divrcnv 14423 . . . . . . . . 9 (1 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
16487, 163mp1i 13 . . . . . . . 8 (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
165 rpreccl 11733 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
166165rpred 11748 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
167166adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ)
16811, 13resubcld 10337 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
169168adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
170 rpaddcl 11730 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑥 + 1) ∈ ℝ+)
17121, 170mpan2 703 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 + 1) ∈ ℝ+)
172171relogcld 24173 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘(𝑥 + 1)) ∈ ℝ)
173172, 13resubcld 10337 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
1747nn0red 11229 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℝ)
175 1red 9934 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → 1 ∈ ℝ)
176 flle 12462 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
1771, 176syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ≤ 𝑥)
178174, 1, 175, 177leadd1dd 10520 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
17910, 171logled 24177 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
180178, 179mpbid 221 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
18111, 172, 13, 180lesub1dd 10522 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
182 logdifbnd 24520 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
183168, 173, 166, 181, 182letrd 10073 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
184183ad2antrl 760 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
185 fllep1 12464 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
1861, 185syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ≤ ((⌊‘𝑥) + 1))
187 logleb 24153 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ ((⌊‘𝑥) + 1) ∈ ℝ+) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
18810, 187mpdan 699 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
189186, 188mpbid 221 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
19011, 13subge0d 10496 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
191189, 190mpbird 246 . . . . . . . . 9 (𝑥 ∈ ℝ+ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
192191ad2antrl 760 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
193161, 162, 164, 167, 169, 184, 192rlimsqz2 14229 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0)
194 rlimo1 14195 . . . . . . 7 ((𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
195193, 194syl 17 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
196 o1mul 14193 . . . . . 6 (((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘𝑓 · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
197160, 195, 196sylancr 694 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘𝑓 · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
198159, 197eqeltrrd 2689 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
199 nnrp 11718 . . . . . . . . 9 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+)
200199ssriv 3572 . . . . . . . 8 ℕ ⊆ ℝ+
201200a1i 11 . . . . . . 7 (⊤ → ℕ ⊆ ℝ+)
202201sselda 3568 . . . . . 6 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
203202, 31syl 17 . . . . 5 ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
204 chpo1ub 24969 . . . . . . . 8 (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
205204a1i 11 . . . . . . 7 (⊤ → (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
206 rerpdivcl 11737 . . . . . . . . 9 (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
20729, 206mpancom 700 . . . . . . . 8 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
208207adantl 481 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
20931adantl 481 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
210 rpreccl 11733 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ+)
211210rpred 11748 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ)
212 chpge0 24652 . . . . . . . . . . 11 (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
21327, 212syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (ψ‘𝑛))
214 logdifbnd 24520 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
21526, 211, 29, 213, 214lemul1ad 10842 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
21627lep1d 10834 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+𝑛 ≤ (𝑛 + 1))
217 logleb 24153 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
21823, 217mpdan 699 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
219216, 218mpbid 221 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
22024, 25subge0d 10496 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
221219, 220mpbird 246 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
22226, 29, 221, 213mulge0d 10483 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
22330, 222absidd 14009 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
224 rpregt0 11722 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
225 divge0 10771 . . . . . . . . . . . 12 ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
22629, 213, 224, 225syl21anc 1317 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((ψ‘𝑛) / 𝑛))
227207, 226absidd 14009 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
22829recnd 9947 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℂ)
229 rpcn 11717 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℂ)
230 rpne0 11724 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ≠ 0)
231228, 229, 230divrec2d 10684 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
232227, 231eqtrd 2644 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
233215, 223, 2323brtr4d 4615 . . . . . . . 8 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
234233ad2antrl 760 . . . . . . 7 ((⊤ ∧ (𝑛 ∈ ℝ+ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
235162, 205, 208, 209, 234o1le 14231 . . . . . 6 (⊤ → (𝑛 ∈ ℝ+ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
236201, 235o1res2 14142 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
237203, 236o1fsum 14386 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
238147, 149, 198, 237o1sub2 14204 . . 3 (⊤ → (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) ∈ 𝑂(1))
239145, 238syl5eqelr 2693 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
240239trud 1484 1 (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ℝ+crp 11708  ...cfz 12197  ..^cfzo 12334  ⌊cfl 12453  abscabs 13822   ⇝𝑟 crli 14064  𝑂(1)co1 14065  Σcsu 14264  logclog 24105  Λcvma 24618  ψcchp 24619 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-o1 14069  df-lo1 14070  df-sum 14265  df-ef 14637  df-e 14638  df-sin 14639  df-cos 14640  df-pi 14642  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-cxp 24108  df-cht 24623  df-vma 24624  df-chp 24625  df-ppi 24626 This theorem is referenced by:  selberg2  25040  selberg3lem2  25047
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